2 Answers
2

I’d say that a full explanation is beyond the scope of an answer here. Section $3$ of this PDF is a fairly gentle but very incomplete start; this PDF goes into much more detail and seems to be about as accessible as a thorough treatment is likely to be, though I’ve only skimmed it.

The explanation may be a bit simpler than seems to be implied by earlier answer. Namely, the hyperreals are a certain quotient of the space $\mathbb{R}^{\mathbb N}$ of sequences of real numbers. More specifically, consider sequences $(x_n)$ that vanish for almost all $n$, and let $I\subset \mathbb{R}^{\mathbb N}$ be a maximal ideal containing all such sequences (the existence of a maximal ideal is proved in undergraduate algebra courses). Then the hyperreals are a quotient field $\mathbb{R}^{\mathbb N}/I$. This is equivalent to a certain construction using an ultrafilter on $\mathbb N$.